Refer to Exercise 7.6. Efron and Tibshirani discuss the following example [84, Ch. 7]. The five-dimensional scores data

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Refer to Exercise 7.6. Efron and Tibshirani discuss the
following example [84, Ch. 7]. The five-dimensional scores data
have a 5 × 5 covariance matrix Σ, with positive eigenvalues λ1 >
··· > λ5. In principal components analysis, θ = λ1 5 j=1 λj
measures the proportion of variance explained by the first
principal component. Let λˆ1 > ··· > λˆ5 be the eigenvalues
of Σ, where ˆ Σ is the MLE of Σ. ˆ Compute the sample estimate ˆθ =
λˆ1 5 j=1 λˆj of θ. Use bootstrap to estimate the bias and
standard error of ˆθ